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Thread: Irreducible Polynomials

  1. #1
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    Irreducible Polynomials

    let a= sqrt(7 + 4 *sqrt(3)) find irr(a,Q)

    now let a= sqrt(7) + i find irr(a,Q)
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  2. #2
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    Quote Originally Posted by Coda202 View Post

    let a= sqrt(7 + 4 *sqrt(3)) find irr(a,Q)
    $\displaystyle \sqrt{7 + 4\sqrt{3}}=2 + \sqrt{3}.$


    now let a= sqrt(7) + i find irr(a,Q)
    $\displaystyle a^4 - 12a^2 + 64=0.$ show that the polynomial $\displaystyle x^4-12x^2+64$ is irreducible over $\displaystyle \mathbb{Z},$ and therefore over $\displaystyle \mathbb{Q}.$ in order to show this you only need to prove that we cannot have

    $\displaystyle x^4-12x^2+64=(x^2+ax+b)(x^2+cx+d),$ for some integers $\displaystyle a,b,c,d.$ that is very easy to verify!
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  3. #3
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    Quote Originally Posted by NonCommAlg View Post
    Show that the polynomial $\displaystyle x^4-12x^2+64$ is irreducible over $\displaystyle \mathbb{Z},$ and therefore over $\displaystyle \mathbb{Q}.$
    How does irreducibile in $\displaystyle \mathbb{Z}$ imply irreducible in $\displaystyle \mathbb{Q}$?
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  4. #4
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    Quote Originally Posted by mathman88 View Post

    How does irreducibile in $\displaystyle \mathbb{Z}$ imply irreducible in $\displaystyle \mathbb{Q}$?
    Gauss's lemma.
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